On the arc-sine laws for Lévy processes
Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t −1 ⨍0 t P 0(X s > 0) ds → c as t → ∞ is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs >0} ds to converge in P 0 law to Fc. Moreover, P 0(Xt > 0) = c for all t...
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| Veröffentlicht in: | Journal of applied probability 1994-03, Vol.31 (1), p.76-89 |
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| container_title | Journal of applied probability |
| container_volume | 31 |
| creator | Getoor, R. K. Sharpe, M. J. |
| description | Let X be a Lévy process on the real line, and let Fc
denote the generalized arcsine law on [0, 1] with parameter c. Then t
−1 ⨍0
t
P
0(X
s
> 0) ds → c as t → ∞ is a necessary and sufficient condition for t
—1 ⨍0
t
1{Xs
>0}
ds to converge in P
0 law to Fc. Moreover, P
0(Xt
> 0) = c for all t > 0 is a necessary and sufficient condition for t
—1 ⨍0
t
1{Xs
>0}
ds under P
0 to have law Fc
for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version. |
| doi_str_mv | 10.2307/3215236 |
| format | Article |
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denote the generalized arcsine law on [0, 1] with parameter c. Then t
−1 ⨍0
t
P
0(X
s
> 0) ds → c as t → ∞ is a necessary and sufficient condition for t
—1 ⨍0
t
1{Xs
>0}
ds to converge in P
0 law to Fc. Moreover, P
0(Xt
> 0) = c for all t > 0 is a necessary and sufficient condition for t
—1 ⨍0
t
1{Xs
>0}
ds under P
0 to have law Fc
for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.</description><identifier>ISSN: 0021-9002</identifier><identifier>EISSN: 1475-6072</identifier><identifier>DOI: 10.2307/3215236</identifier><identifier>CODEN: JPRBAM</identifier><language>eng</language><publisher>Cambridge, UK: Cambridge University Press</publisher><subject>Continuous functions ; Density ; Exact sciences and technology ; Markov processes ; Mathematical models ; Mathematical theorems ; Mathematics ; Poisson process ; Probability ; Probability and statistics ; Probability theory ; Probability theory and stochastic processes ; Random variables ; Random walk ; Random walk theory ; Research Papers ; Sciences and techniques of general use ; Semigroups ; Spectral index ; Stationary ; Studies ; Utility functions</subject><ispartof>Journal of applied probability, 1994-03, Vol.31 (1), p.76-89</ispartof><rights>Copyright © Applied Probability Trust 1994</rights><rights>Copyright 1994 Applied Probability Trust</rights><rights>1994 INIST-CNRS</rights><rights>Copyright Applied Probability Trust Mar 1994</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c274t-9b181b18e311122f0f48d8bb1ccdcb3fa20b6c040ecc9afe4ff9a44526a4a8c83</citedby><cites>FETCH-LOGICAL-c274t-9b181b18e311122f0f48d8bb1ccdcb3fa20b6c040ecc9afe4ff9a44526a4a8c83</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/3215236$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/3215236$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,780,784,803,832,27924,27925,58017,58021,58250,58254</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=4054232$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Getoor, R. K.</creatorcontrib><creatorcontrib>Sharpe, M. J.</creatorcontrib><title>On the arc-sine laws for Lévy processes</title><title>Journal of applied probability</title><addtitle>Journal of Applied Probability</addtitle><description>Let X be a Lévy process on the real line, and let Fc
denote the generalized arcsine law on [0, 1] with parameter c. Then t
−1 ⨍0
t
P
0(X
s
> 0) ds → c as t → ∞ is a necessary and sufficient condition for t
—1 ⨍0
t
1{Xs
>0}
ds to converge in P
0 law to Fc. Moreover, P
0(Xt
> 0) = c for all t > 0 is a necessary and sufficient condition for t
—1 ⨍0
t
1{Xs
>0}
ds under P
0 to have law Fc
for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.</description><subject>Continuous functions</subject><subject>Density</subject><subject>Exact sciences and technology</subject><subject>Markov processes</subject><subject>Mathematical models</subject><subject>Mathematical theorems</subject><subject>Mathematics</subject><subject>Poisson process</subject><subject>Probability</subject><subject>Probability and statistics</subject><subject>Probability theory</subject><subject>Probability theory and stochastic processes</subject><subject>Random variables</subject><subject>Random walk</subject><subject>Random walk theory</subject><subject>Research Papers</subject><subject>Sciences and techniques of general use</subject><subject>Semigroups</subject><subject>Spectral index</subject><subject>Stationary</subject><subject>Studies</subject><subject>Utility functions</subject><issn>0021-9002</issn><issn>1475-6072</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1994</creationdate><recordtype>article</recordtype><recordid>eNp9kE1Lw0AQhhdRsFbxLwQR1EN0dnfydZTiFxR60XOYbHc1IU3qTmrpT_J3-MeMNFjw4GFmLg_PO7xCnEq4VhqSG61kpHS8J0YSkyiMIVH7YgSgZJj1-1AcMVcAEqMsGYnLWRN0bzYgb0IuGxvUtObAtT6Yfn1-bIKlb41ltnwsDhzVbE-GOxYv93fPk8dwOnt4mtxOQ6MS7MKskKnsx2oppVIOHKbztCikMXNTaEcKitgAgjUmI2fRuYwQIxUTUmpSPRZnW2-f_L6y3OVVu_JNH5krjRlIjbqHLraQ8S2zty5f-nJBfpNLyH9ayIcWevJ80BEbqp2nxpT8iyNEqLTaYRV3rf_HdjXk0qLw5fzV7r77y34Dqq9yKA</recordid><startdate>19940301</startdate><enddate>19940301</enddate><creator>Getoor, R. K.</creator><creator>Sharpe, M. J.</creator><general>Cambridge University Press</general><general>Applied Probability Trust</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>H8D</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>19940301</creationdate><title>On the arc-sine laws for Lévy processes</title><author>Getoor, R. K. ; Sharpe, M. J.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c274t-9b181b18e311122f0f48d8bb1ccdcb3fa20b6c040ecc9afe4ff9a44526a4a8c83</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1994</creationdate><topic>Continuous functions</topic><topic>Density</topic><topic>Exact sciences and technology</topic><topic>Markov processes</topic><topic>Mathematical models</topic><topic>Mathematical theorems</topic><topic>Mathematics</topic><topic>Poisson process</topic><topic>Probability</topic><topic>Probability and statistics</topic><topic>Probability theory</topic><topic>Probability theory and stochastic processes</topic><topic>Random variables</topic><topic>Random walk</topic><topic>Random walk theory</topic><topic>Research Papers</topic><topic>Sciences and techniques of general use</topic><topic>Semigroups</topic><topic>Spectral index</topic><topic>Stationary</topic><topic>Studies</topic><topic>Utility functions</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Getoor, R. K.</creatorcontrib><creatorcontrib>Sharpe, M. J.</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>Aerospace Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Journal of applied probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Getoor, R. K.</au><au>Sharpe, M. J.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the arc-sine laws for Lévy processes</atitle><jtitle>Journal of applied probability</jtitle><addtitle>Journal of Applied Probability</addtitle><date>1994-03-01</date><risdate>1994</risdate><volume>31</volume><issue>1</issue><spage>76</spage><epage>89</epage><pages>76-89</pages><issn>0021-9002</issn><eissn>1475-6072</eissn><coden>JPRBAM</coden><abstract>Let X be a Lévy process on the real line, and let Fc
denote the generalized arcsine law on [0, 1] with parameter c. Then t
−1 ⨍0
t
P
0(X
s
> 0) ds → c as t → ∞ is a necessary and sufficient condition for t
—1 ⨍0
t
1{Xs
>0}
ds to converge in P
0 law to Fc. Moreover, P
0(Xt
> 0) = c for all t > 0 is a necessary and sufficient condition for t
—1 ⨍0
t
1{Xs
>0}
ds under P
0 to have law Fc
for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.</abstract><cop>Cambridge, UK</cop><pub>Cambridge University Press</pub><doi>10.2307/3215236</doi><tpages>14</tpages></addata></record> |
| fulltext | fulltext |
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| ispartof | Journal of applied probability, 1994-03, Vol.31 (1), p.76-89 |
| issn | 0021-9002 1475-6072 |
| language | eng |
| recordid | cdi_proquest_journals_234901343 |
| source | JSTOR Mathematics & Statistics; JSTOR Archive Collection A-Z Listing |
| subjects | Continuous functions Density Exact sciences and technology Markov processes Mathematical models Mathematical theorems Mathematics Poisson process Probability Probability and statistics Probability theory Probability theory and stochastic processes Random variables Random walk Random walk theory Research Papers Sciences and techniques of general use Semigroups Spectral index Stationary Studies Utility functions |
| title | On the arc-sine laws for Lévy processes |
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